Enumeration of Type D Permutations with Alternating Runs

Abstract

In the study of enumeration polynomials of signed permutations of rank n, which is known as a Coxeter group of type B, Chow and Ma found that alternating runs of up signed permutations are closely related to peaks and valleys of these permutations. Notice that even-singed permutations of rank n, which is also called a Coxeter group of type D, forms a subgroup of signed permutations of index 2, we study the number of type D permutations according to alternating runs and consider how alternating runs connect with peaks and valleys. We find in this paper that the generating function of alternating runs of up even-signed permutations can be expressed as those generating functions of peaks and valleys of up even-signed permutations, which partially provide an affirmative answer to a conjecture by Chow and Ma. Additionally, we establish a recurrence for the generating function of alternating runs and an identity on alternating runs of type D permutations.